CONSTRUCTION OF SELF-DUAL CODES OVER Z6, Z8 AND Z10

• Main Author: Ahsanunnisa, Maryam
• Format: Theses
• Language: Indonesia
• Subjects: Ilmu alam dan matematika
• Online Access: https://digilib.itb.ac.id/gdl/view/37136
• Institution: Institut Teknologi Bandung

Let m > 0 be even and let Zm be the ring of integers modulo m: A linear code C of length n over Zm is Zm-submodule of Zn m: Euclidean weight of x = (x1;x2; ;xn) 2 Zn m; denoted by wE(x); is defined by wE(x) = nå i=1 minfx2 i ; (2k????xi)2g: Minimum Euclidean weight of the linear code C; denoted by dE(C); is the smallest Euclidean weight of non-zero vectors x 2 C: For v;w 2 Zn m; an inner-product of v and w is defined by [v;w] = nå i=1 viwi: Dual code C? of C consists of vectors in Zn m which are orthogonal to all vectors in C with respect to the above inner-product. The code C is called self-dual if C? =C: The self-dual codes is called Type II if the Euclidean weights of all codewords in C are divisible by 2m and is called Type I otherwise. Gulliver and Harada [3] provided the upper bound for Type I and Type II codes over Z6; Z8; and Z10. The Type I (resp. II) code is called extremal if its minimum Euclidean distance attains the upper bound. In this thesis, we review the construction methods of self-dual codes over Z6, Z8 and Z10 as introduced by Dougherty et.al. [2] and Lee and Lee [5]. Moreover, we also provided examples of several extremal codes of modest lengths obtained by the methods.